Idea

Category theory frequently allows one to give precise and useful formalized meanings to “everyday” terms, at least terms used everyday by practicing mathematicians.

It was indeed introduced originally in order to formalize the use of the notion “natural” in mathematics. Another frequently recurring pair of terms in math are “extra structure” and “extra properties”, to which we add the more general concept of “extra stuff”. In discussion among Jim Dolan, John Baez and Toby Bartels, the following useful formalization of these concepts in category theoretic terms was established.

and that every 1-functor is essentially kk-surjective for all k≥3k \geq 3.

So the above says for a functor F:C→DF : C \to D:

If it is …

then it …

but it …

essentially (k≥0)(k \geq 0)-surjective

forgets nothing

remembers everything

essentially (k≥1)(k \geq 1)-surjective

forgets only properties

remembers at least stuff and structure

essentially (k≥2)(k \geq 2)-surjective

forgets at most structure

remembers at least stuff

essentially (k≥3)(k \geq 3)-surjective

may forget everything

may remember nothing

It is worth noting that this formalism captures the intuition of how “stuff”, “structure”, and “properties” are expected to be related:

stuff may be equipped with structure;

structure may have (be equipped with) properties.

The 33-way breakdown looks like this:

If it is …

then it …

but it …

essentially (k≠0)(k \ne 0)-surjective

forgets only properties

remembers at least stuff and structure

essentially (k≠1)(k \ne 1)-surjective

forgets purely structure

remembers at least stuff and properties

essentially (k≠2)(k \ne 2)-surjective

forgets purely stuff

remembers at least structure and properties

This formalism does not capture the intuition so well, and in fact the ‘properties’ (and ‘structure’) remembered by a functor that forgets purely structure (or purely stuff) may not match what one expects.

See also the examples below.

Generalization to higher groupoids

The formulation in terms of kk-surjectivity induces an immediate generalization of the notions of stuff, structure and properties to the context of infinity-groupoids. Baez’s students speak of “2-stuff,” “3-stuff,” and so on. Of course, structure and properties can then be called 0-stuff and (−1)(-1)-stuff, respectively.

Generalization to categories and higher categories

The theory is easiest when restricted to groupoids as above; for categories, there are several ways to go. One is to keep the definition as phrased above (a functor between categories forgets only properties if it is fully faithful, forgets at most structure if it is faithful, etc.). Another is to apply the above definition instead to the functor between the underlying groupoids of the categories in question.

To tell the difference, ask yourself whether the difference between a monoid and a semigroup is the structure of being equipped with an identity element or only the property that an identity element exists. Note that an identity element, if it exists, must be unique and must be preserved by semigroup isomorphisms and by monoid homomorphisms but not by semigroup homomorphisms.

A third option is to define a new notion: a functor forgets at most property-like structure if it is pseudomonic. This means that (1) the functor is faithful and (2) its induced functor between underlying groupoids is fully faithful. Intuitively, property-like structure can be described as consisting of “properties which need not automatically be preserved by morphisms” or “structure which, if it exists, is uniquely determined.”

Property-like structure becomes much more prevalent for higher categories. For example, the forgetful functor from the 2-category of cartesian monoidal categories (and product-preserving functors) to Cat is essentially (k≥2)(k\ge 2)-surjective, and its induced functor between 2-groupoids is essentially (k≥1)(k\ge 1)-surjective; thus it forgets property-like structure. See also lax-idempotent 2-monad.

Note that property-like structure is known in traditional logic as categorical structure. Obviously, this term can be confusing in categorial logic!

Specifically, given a functor F:C→DF: C \to D, let the 1-image1imF1 im F of FF be the category whose objects are objects of CC and whose morphisms x→yx \to y are morphisms F(x)→F(y)F(x) \to F(y) in DD; let the 2-image2imF2 im F of FF be the category whose objects are objects of CC and whose morphisms x→yx \to y are morphisms b:F(x)→F(y)b: F(x) \to F(y) in DD such that b=F(a)b = F(a) for some a:x→ya: x \to y in CC. (So the only difference bewteen 2imF2 im F and CC itself is equality of morphisms.) If you want to be complete, call CC itself the 3-image of FF and DD the 0-image.

Examples

The classical categories of sets with extra structure

The classical examples are the forgetful functors to Set that define the classical categories such as Top, Grp, Vect, etc. All these categories are categories of sets equipped with extra structure (e.g. with a topology, with a group structure, etc). Accordingly the obvious functors to Set are

faithful

not full.

Hence indeed, by the above yoga, they forget this extra structure but remember the stuff in question (the underlying set).

More examples

The embedding of abelian groups into all groups, F:F :Ab→\toGrp is faithful and full, but not essentially surjective. Hence it should remember stuff and structure but forget properties. Indeed, the property it forgets is the property “is abelian” which is a property of the group structure sitting on the underlying set of a group. Hence the sequence of functors

Ab→Grp→Set→pt
Ab \to Grp \to Set \to pt

(with pt the terminal category) successively forgets first a property (being abelian) then structure (the group structure on a set) then stuff (the underlying set).

Notice that the order here is backwards from the automatic factorisation given by the 33-way factorisation system described above. (And in fact, the structure forgotten here is not ‘pure’; Grp→SetGrp \to Set is not essentially surjective.) Indeed, the above factorisation is arbitrary; it comes from seeing an abelian group as a group with an extra property and a group as a set with extra structure, but one may view things differently (for example, that an abelian group is a monoid with extra property, or a set with two group structures that are related as in the Eckmann-Hilton argument).

because the original functor Ab→ptAb \to \pt is already essentially surjective and full. In other words, from the perspective of pt\pt, an abelian group is simply extra stuff.

More interestingly, we can factor the forgetful functor Ab→SetAb \to Set:

Ab→Ab→Set∖{∅}→Set
Ab \to Ab \to Set \setminus \{\empty\} \to Set

Here, the first part is trivial because Ab→SetAb \to Set is faithful. The category Set∖{∅}Set \setminus \{\empty\} is the category of inhabited sets, that is the category of sets that are capable of being equipped with the structure of an abelian group. So from the point of view of its underlying set, an abelian group is a set with the property that it is inhabited and the structure of an abelian group but no additional stuff.

For something interesting at every level, take the functor Ab×Ab→SetAb \times Ab \to Set that takes the underlying set of the first abelian group. This factors as follows:

So a pair of abelian groups (from the perspective of the underlying set of the first one) consists of the property that the set is inhabited, then the structure of an abelian group on that set, and finally extra stuff consisting of the entire second group.

Logical interpretation

In logic a property PP is given by a predicate, which we may think of as an operation that takes a thing xx to the truth value of the statement “xx has property PP”. Note that a truth value is a (−1)(-1)-groupoid; we get structure and stuff by replacing this with a set (a 00-groupoid) or a groupoid (a 11-groupoid) and we get nn-stuff by replacing this with an nn-groupoid.

Now, if a non-evil property of objects of a category CC holds for some object xx, then it must hold for any object isomorphic to xx. That is, the predicate defining that property is actually a functor from the coreCisoC_iso of CC to the groupoid TVisoTV_iso of truth values. Given such a predicate functor PP, it's immediate how to define a full subcategoryCPC_P of CC consisting of those objects with the property; the inclusion functorCP↪CC_P \hookrightarrow C is fully faithful, as it should be for extra property. Conversely, given a fully faithful functor F:D→CF: D \to C, define a non-evil property of objects of CC as follows: an object xx of CC has the property if there is some object aa of DD such that x≅F(a)x \cong F(a). If you apply this to CP↪CC_P \hookrightarrow C, then you get the predicate PP back; if you start with an arbitrary fully faithful F:D→CF: D \to C, define a predicate PP, and then form CFC_F, you'll find that CFC_F and DD are equivalent, even as bundles over CC.

Similarly, any non-evil structure on objects of CC is given by a functor from CisoC_iso to the groupoid SetisoSet_iso of sets. Given such a functor PP, let CPC_P be the category of elements of PP, which comes with a faithful functor from CPC_P to CC. Conversely, given any faithful functor F:D→CF: D \to C and an object xx of CC, let P(x)P(x) be the essential fiber of FF over xx, which (because FF is faithful) is a discrete category and hence (equivalent to) a set. These operations are also invertible, up to equivalence.

Next, any non-evil stuff on objects of CC is given by a functor from CisoC_iso to GrpdisoGrpd_iso. Here GrpdisoGrpd_iso should be taken to be the 22-groupoid whose objects are groupoids, whose morphisms are equivalences, and whose 22-morphisms are natural isomorphisms; similarly, the functor P:Ciso→GrpdisoP: C_iso \to Grpd_iso should be taken in the weakest sense (often called a pseudofunctor). Then the Grothendieck construction turns PP into a category CPC_P equipped with a functor to CC; again, the essential fiber converts any functor F:D→CF: D \to C into such a PP (although really you must take the core of the essential fiber to get a groupoid).

If CC is a mere 11-category, then any P:Ciso→2GrpdisoP: C_iso \to 2 Grpd_iso is equivalent to some P:Ciso→GrpdisoP: C_iso \to Grpd_iso, but in general we need to consider P:Ciso→nGrpdisoP: C_iso \to n Grpd_iso or P:Ciso→∞GrpdisoP: C_iso \to \infty Grpd_iso to study higher forms of nn-stuff.

Lest we forget, to be even more simple than an extra property, the groupoid of (−2)(-2)-groupoids is the pointptpt, and there is exactly one functor PP from any CisoC_iso to ptpt, corresponding to the unique (up to equivalence) category equivalent to CC.